Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.

51 Results for the subject "Trigonometry":

Publisher SummaryThis chapter discusses the circular functions of trigonometry. The word trigonometryderives from the Greek meaning measurement of triangles. The conventional approach to the subject matter of trigonometry deals with relationships between the sides and angles of a triangle, reflecting the important applications of trigonometry in such fields as navigation and surveying. The modern approach is to deal with trigonometry in terms of functional relationships. The trigonometric functions can be viewed as functions whose domains are angles or as functions whose domains are real numbers, in which case they are often referred to as the circular functions. The latter approach more clearly demonstrates the utility of the function concept.

BERNARD KOLMAN Publication date: 1981/01/01AbstractThe main conclusion of this paper is that the Bell–Wigner–Accardi theory of quantum probabilities in spin systems may be placed within the general operator trigonometry developed independently by this author about 30 years ago. The use of the Grammian from the operator trigonometry simplifies and clarifies the analysis of Wigner. A general triangle inequality from the operator trigonometry clarifies and generalizes the analysis of Accardi. The statistical meaning of the complex numbers in quantum mechanics is seen to be that of the natural geometry of the operator trigonometry. A new connection of the operator trigonometry to CP symmetry violation is established.

Karl Gustafson Publication date: 2001/11/12AbstractTrigonometry is an essential part of engineering mathematics. For example, in robotics, trigonometry can be useful in calculating the positions of robotic arms, rotations as well as other quantities. In addition, trigonometrical functions are also intrinsically related to complex numbers. This chapter introduces the fundamentals of trigonometrical functions.

Xin-She Yang Publication date: 2017/01/01AbstractThe operator trigonometry of symmetric positive definite (SPD) matrices is extended to arbitrary invertible matrices A and to arbitrary invertible bounded operators A on a Hilbert space. Some background and motivation for these results is provided.

Karl Gustafson Publication date: 2000/11/01Publisher SummaryThis chapter discusses trigonometry. Trigonometry is the branch of mathematics that deals with the measurement of triangles. Trigonometry helps in finding the unknown parts of triangles by arithmetical processes. For this reason it is constantly used in surveying, mechanics, navigation, engineering, physics, and astronomy. The chapter explains that the angle of elevation or depression of an object is the angle made between a line from the eye to the object and a horizontal line in the same vertical plane. If the object is above the horizontal line, it makes an angle of elevation; if below the horizontal line it makes an angle of depression.

Abraham Sperling Publication date: 1981/01/01Publisher SummaryThis chapter provides an overview of trigonometry, which means triangle measurement. Any two angles that sum to a right angle are called complementary angles. An angle smaller than a right angle is called an acute angle, and an angle larger than a right angle is called an obtuse angle. Angles add just like other sets of numbers. Any two angles which, when added together, make a straight line are called supplementary angles. A vector is a representation of both a quantity and a direction. The sum of forces at any place and at any time must equal to zero. Laws of trigonometry and periodicity are also discussed in this chapter.

Archibald L. Fripp Publication date: 2003/01/01Publisher SummaryThis chapter provides an overview of trigonometry of the circular functions. From the definitions of trigonometry, it is understood that if t is real and W (t) = (x, y), then sin t = y, cos t = x, and tan . The domain of tan t is restricted so that x ≠ 0, that is, D: tan t = all reals except , where n is an integer. If a function, f (−x) = f (x), f is said to be even; on the other hand, if f (−x) = −f (−x), then f is said to be odd. A periodic function is one in which the function values repeat at a constant interval. The period of sin t and cos t is 2π. Both sin t and cos t show their periodic or wave nature by repeating y values at 2π intervals. The tangent function is also periodic, but its period is π. Composites of the trigonometric functions—for example, (1 − cos t) and (sin t + cos t)—can be graphed by adding the separate parts.

James W. Snow Publication date: 1981/01/01Publisher SummaryDuring the first decades of the 19th century, duality began to be considered as a mathematical subject. It was dealt with in a variety of ways, associated respectively with the names of Poncelet, Gergonne, Plücker, and others. This chapter focuses on the approach to duality developed by Joseph Diaz Gergonne as it emerged in a series of works that appeared from 1810. Gergonne was interested in the fact that various domains of geometry—the theory of polyhedra, spherical trigonometry, some parts of plane and solid geometry—present a common phenomenon. They can be presented in such a way that their theorems and proofs are joined in couples, the members of which correspond to one another by a systematic linguistic translation. For all these domains, including spherical trigonometry, he inserted such presentations in his journal, although it was not until the mid-1820s that he drew explicit conclusions about the essence of this phenomenon of duality. The chapter also pr *Read more...*

Publisher SummaryThis chapter discusses the essential elements from algebra, plane geometry, and analytic geometry in study of trigonometry. It is necessary in study of trigonometry to distinguish equations from identities. Equation is an equality that is true only for certain values of the unknown. An identity is an equality that is true for all values of the unknown for which both members are defined. The concept of an inverse function is also needed in study of trigonometry. The concept of an angle and its measure in degrees, minutes, and seconds as given in plane geometry is extended in trigonometry. A new measure of an angle is used as follows: (1) consider a circle of any radius r and with central angle θ such that the radii intercept an arc S on the circumference equal in length to the radius of the circle; and (2) the angle θ is then said to be an angle of one radian and has a measure of one radian. The central angle has a measure given by the length of arc. In analytic geome *Read more...*

No abstract

Andrew S. Glassner Publication date: 1990/01/01No abstract

John Bird Publication date: 2008/01/01No abstract

D.W. HILDER Publication date: 1966/01/01No abstract

Jim Ver Hague Publication date: 2006/01/01AbstractWe present a method for measurement of thickness of transparent oil film on water surface based on laser trigonometry. With an oblique incident mode of single-point laser triangulation ranging system, laser light is incident on the upper and lower surfaces of the oil film being measured and an ellipse light spot is formed on the upper and lower surfaces of the oil film. The two light spots are imaged on an image plane CCD by an imaging lens and the image spot is formed and stored in a computer. The thickness of oil film being measured can be obtained by displacement of the image spot and the configuration parameter of the imaging system. The experiment is conducted using edible peanut oil and diesel oil. The research results show that the method presented in this paper is feasible and applicable to dynamic on-line measurement of oil film thickness of oil spill on sea surface.

Lü Qieni Publication date: 2011/01/01No abstract

Ken Turkowski Publication date: 1990/01/01No abstract

Patrick D. Barry Publication date: 2016/01/01AbstractWe inquire into an operator-trigonometric analysis of certain multi-asset financial pricing models. Our goal is to provide a new geometric point of view for the understanding and analysis of such financial instruments. Among those instruments which we examine are quantos for currency hedging, spread options for multi-asset pricing, portfolio rebalancing under stochastic interest rates, Black–Scholes volatility models, and risk measures.

Karl Gustafson Publication date: 2010/02/01AbstractDue to the lack of exact quantitative information or the difficulty associated with obtaining or processing such information, qualitative spatial knowledge representation and reasoning often become an essential means for solving spatial constraint problems as found in science and engineering. This paper presents a computational approach to representing and reasoning about spatial constraints in two-dimensional Euclidean space, where the a priori spatial information is not precisely expressed in quantitative terms. The spatial quantities considered in this work are qualitative distances and qualitative orientation angles. Here, we explicitly define the semantics of these quantities and thereafter formulate a representation of qualitative trigonometry (QTRIG). The resulting QTRIG formalism provides the necessary inference rules for qualitative spatial reasoning. In the paper, we illustrate how the QTRIG relationships can be employed in generating qualitative spatial descriptions *Read more...*

AbstractIn order to construct an extension of the complex numbers, we consider an n-dimensional commutative algebra generated by the n vectors 1, e, ..., en−1 where the fundamental element satisfies the basic relation en = −1. These spaces can be classified according to the values of n: prime number, power of a prime number, general number. The question of the invertibility leads to the definition of a pseudo-norm for which the triangle inequality is not satisfied (the n = 1, 2 cases excepted). When one tries to pass from the polar form the cartesian one, one obtains functions generalizing the usual circular and hyperbolic functions and their inverse. The extended sine and hyperbolic sine functions thus constructed satisfy a determinantal-type relation and they lay the foundation of a new trigonometry for which summation and derivative formulas are given. An extended 2π quantity is defined as the periodicity of the generalized circular functions. This formalism is applied to solve *Read more...*

ObjectivesTo retrospectively assess the usefulness of the measurements on preoperative computed tomography (CT) of patients with urinary stone disease for planning the access site using vertical angulation of the C-arm.MethodsOf the patients who underwent percutaneous nephrolithotomy from November 2001 to October 2006, 41 patients with superior calix access had undergone preoperative CT. The depth of the target stone (y) and the vertical distance from that point to the first rib free slice (x) were measured on CT. The limit of the ratio of x over y was accepted as 0.58, with ratios below that indicating that infracostal access could be achieved by vertical angulation of the C-arm.ResultsWe achieved an approach to the superior calix through an infracostal access in 28 patients. The preoperative trigonometric study on CT predicted 24 of them. The stone-free rate was 92.6%, and no chest-related complications developed.ConclusionsSimple trigonometry on CT of the patients with complex stone *Read more...*